

The issue of regularity for systems of complex flows based on the NavierStokes equations


Wojciech Zajączkowski (IMPAN) and Gregory A. Seregin (Mathematical Institute University of Oxford)


The project is carried out by Magdalena Bogdańska at IM PAN
 
This project is devoted to the problem of regularity of weak solutions to the NavierStokes equations. We can distinguish the following directions:
(1) formulation of Serrin type conditions for some special solutions
(2) problem of existence of global regular solutions to the NavierStokes equations which remain close to either the twodimensional
solutions, or to the axially symmetric solutions, or to the helicoidal solutions.
(3) development of new techniques as Besov spaces, BMO spaces with an essential use of the harmonic analysis
(4) examining of the existence of global regular solutions of the NavierStokes equations coupled with the heat equation, equations
describing internal properties of the fluid, and also the same for the nonhomogeneous NavierStokes equations and magnetohydrodynamics.
Examining the existence of regular solutions to the NavierStokes equations which are close to the axially symmetric solutions we need the
technique of weighted Sobolev spaces. Therefore we need the solvability of boundary and initialboundary value problems for the
Laplace equation, the heat equations and the Stokes system in the weighted Sobolev spaces.
[1] W. Zajączkowski (2008) Zapiski Nauchn.Sem POMI.
[2] J. Y. Chemin, I. Gallagher, M.Paicu, (2008) arXiv:~0807.1265v1
[4] G. Seregin, (2006) arXiv:0607537v2




